9
votes
Accepted
Application of Poisson process in economic modelling
Most of the literature on "Strategic experimentation" (or Bandits) uses Poisson processes. Here players can use either a risky or safe arm and one of them generates a fixed stream of payoffs (usually ...
- 1,682
9
votes
Accepted
Complete Markets in Continuous Time
I am the last person that should be answering continuous time questions like these, but if there's no one else I guess I'll give it a shot. (Any correction of my dimly remembered continuous-time ...
- 3,712
9
votes
References to learn continuous-time dynamic programming
For continuous-time stochastic dynamic programming, the small, nontechnical Art of Smooth Pasting by Dixit is a wonderful option. It does a very effective job of conveying the basic intuition.
Stokey'...
- 3,712
6
votes
References to learn continuous-time dynamic programming
Dynamic Programming & Optimal Control by Bertsekas
Introduction to Modern Economic Growth by Acemoglu
The Acemoglu book, even though it specializes in growth theory, does a very good job ...
- 714
5
votes
Accepted
From Discrete to Continuous time: Total Differential
You can separate your function in three terms by writing
\begin{align}
& v(c_{t+\Delta},u_{t+\Delta},t+\Delta)-v(c_t,u_t,t) = \\
& v(c_{t+\Delta},u_{t+\Delta},t+\Delta)-v(c_t,u_{t+\Delta},t+\...
- 3,202
5
votes
Complete Markets in Continuous Time
I've been meaning to post this for a long time. I came across this and thought it could add some insight. This example is from "Financial Asset Pricing Theory" by Munk.
Consider the ...
- 9,155
5
votes
References to learn continuous-time dynamic programming
I think Kamien and Schwartz's Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management is pretty well known.
- 9,155
4
votes
Compute evolution of a distribution over time
Here's my best guess. I haven't checked to thoroughly if this is right, but maybe it will help.
Evolution of population density
I understand the model as follows. $f(a,t)$
is the density of people ...
- 9,155
4
votes
References to learn continuous-time dynamic programming
Controlled Markov Processes and Viscosity Solutions by Fleming and Soner includes a number of applications to Finance and Differential Games.
- 278
3
votes
Accepted
Optimal consumption in Merton-like portfolio choice model with constant wage
$\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\E}{\mathbb{E}}
%short command for inseting abbreviated "such that" in a ...
- 9,155
3
votes
Application of Poisson process in economic modelling
Klette and Kortum (2004) develop a parsimonious model of innovating firms rich enough to confront firm-level evidence. It captures the dynamics of individual heterogenous firms, describes the behavior ...
- 6,662
3
votes
References to learn continuous-time dynamic programming
A really nice methodology for approximating the HJB is the upwind scheme, which I learnt quite quickly using Ben Moll et al's notes and codes
The examples are continuous time versions of familiar ...
- 10.4k
2
votes
Solow model, time and steady state
In the model with technological progress the capital per effective worker remains constant, implies that capital per worker grows at the rate of exogenous rate of technological progress. See Barro and ...
2
votes
Matching problem in continuous time
Yes, it is correct. You can (for instance) write a Taylor expansion:
\begin{align*}
[1-(1-\frac{x}{un})^n]^x & = [1-e^{n ln(1-\frac{x}{un})}]^x \\
& = [1-e^{n (-\frac{x}{un} + o(\frac{1}{n}))}]...
- 3,202
2
votes
Matching problem in continuous time
An alternative approximating approach you could use as as check might be to say there are $X$ job offers in total and $u$ unemployed.
So the probability that an individual does not get a particular ...
- 4,694
2
votes
Accepted
Update of value function in continuous time - HJB
You iterate towards a fixed point, so you want to reach a situation where plugging in your current iterated value produces itself. Now using your notation, we are told that we should calculate
$$V_{n+...
- 31.9k
2
votes
Accepted
Computing the continuous time survival rate
I think the step
"...where $P(a, t) = exp(-d(a,t)\Delta)$ is the discrete time analogue of $d(a,t)$..."
is the problem.
In continuous time I guess we have
$$\dot m(a,t) = -d(a,t)m(a,t) \implies ...
- 31.9k
2
votes
References to learn continuous-time dynamic programming
Applied Intertemporal Optimization by Klaus Wälde is a very very nice book, even for those who are not really familiar with mathematics.
The book treats deterministic and stochastic models, both in ...
- 2,095
2
votes
Solving a HJB with a probability to transit to a new state
I would leave this as a comment but I cant. You are on the right track.
Once you know $V_2(k)$ then you can plug that into to the first hjb and solve.
To solve for $V_2$ you need to find the optimal ...
- 151
2
votes
Accepted
Help understanding expression for continuous discounting
From time $0$ to time $t$ the profit flow is constant $\Delta$. The discounted value of the profit flow of any instance $s$ is $\Delta \cdot e^{-r s}$. To get the total discounted profit we will need ...
- 26.7k
1
vote
Multiple solutions to an HJB, how to pin down the optimal "viscosity" solution?
Answer to Q1:
If we re-write FONC as a function of $a$: $u'(c(a))-V_{a} =0$
Differentiate wrt $a$ (as in Walde 2010): $u''(c(a)) c'(a) -V_{aa} =0$
We know from SOSC that $u''(c(a))<0$.
If we ...
- 310
1
vote
What is the relationship between the HJB and "Hamiltonian"? Why is the Hamiltonian H(p) inside the HJB?
Since you mention Walton, here is something from his notes page 111 Definition 4:
The Hamiltonian and is defined
$H(t,x,\rho) := min _v \{c(t,x,v) - \rho v\}$
Notice the analogue to the Hamiltonian ...
- 721
1
vote
Relationship of continuous and discrete time models
Poisson process can be interpreted as a continuous case of Bernoulli process.
Taking your example, consider that the buyer is consuming the good in batches in fixed intervals of time with probability $...
- 1,630
1
vote
He, Krishnamurthy (2013)
The paper is not trying to say that equation (10) is derived from equation (8). Equation (8) tells us how household makes its optimal consumption and 'saving' decision (it gives us demand for ...
- 43.6k
1
vote
Stochastic growth in continuous time
More of a comment:
There should be an expectation operator in the statement of the problem, otherwise problem doesn't make sense.
That "...the deterministic and stochastic value function must be the ...
- 2,569
1
vote
Accepted
Intuition of the Kolmogorov Equations
I will try to answer to your last question. I did not read the paper but in models with higher dimensions, it is always difficult to find an analytical solution.
If there exists an analytical ...
- 2,095
1
vote
Complete Markets in Continuous Time
Mathematically, market completeness in continuous-time models does not follow from discrete-time heuristics.
In discrete-time, market completeness replies on only linear algebraic considerations.
...
- 2,569
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